Problem: How many numbers between $1$ and $100$ (inclusive) are divisible by $3$ or $4$ ?
There are $33$ numbers divisible by $3$ between $1$ and $100$, and $25$ numbers divisible by $4$ between $1$ and $100$. So, you might think there are $33 + 25 = 58$ numbers divisible by one or the other, but this is overcounting something. We're counting every number which is divisible by both $3$ and $4$ twice. So, for example, $12$ is counted once as a number divisible by $3$, and then again as a number divisible by $4$. So, we need to count how many numbers are divisible by both $3$ and $4$ and subtract this from what we had before. Being divisible by both $3$ and $4$ is the same thing as being divisible by $12$, so there are $8$ numbers between $1$ and $100$ divisible by both. Subtracting, there are $58 - 8 = 50$ numbers divisible by $3$ or $4$.